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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dtbtrs (f07ve)

## Purpose

nag_lapack_dtbtrs (f07ve) solves a real triangular band system of linear equations with multiple right-hand sides, AX = B$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$.

## Syntax

[b, info] = f07ve(uplo, trans, diag, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dtbtrs(uplo, trans, diag, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dtbtrs (f07ve) solves a real triangular band system of linear equations AX = B$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether A$A$ is upper or lower triangular.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A$A$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A$A$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
The equations are of the form AX = B$AX=B$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
The equations are of the form ATX = B${A}^{\mathrm{T}}X=B$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
3:     diag – string (length ≥ 1)
Indicates whether A$A$ is a nonunit or unit triangular matrix.
diag = 'N'${\mathbf{diag}}=\text{'N'}$
A$A$ is a nonunit triangular matrix.
diag = 'U'${\mathbf{diag}}=\text{'U'}$
A$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1$1$.
Constraint: diag = 'N'${\mathbf{diag}}=\text{'N'}$ or 'U'$\text{'U'}$.
4:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals of the matrix A$A$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
5:     ab(ldab, : $:$) – double array
The first dimension of the array ab must be at least kd + 1${\mathbf{kd}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ triangular band matrix A$A$.
The matrix is stored in rows 1$1$ to kd + 1${k}_{d}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ij${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(1 + ij,j)​ for ​jimin (n,j + kd).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If diag = 'U'${\mathbf{diag}}=\text{'U'}$, the diagonal elements of A$A$ are assumed to be 1$1$, and are not referenced.
6:     b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ right-hand side matrix B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the array ab.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
r$r$, the number of right-hand sides.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldab ldb

### Output Parameters

1:     b(ldb, : $:$) – double array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ solution matrix X$X$.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: trans, 3: diag, 4: n, 5: kd, 6: nrhs_p, 7: ab, 8: ldab, 9: b, 10: ldb, 11: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, a(i,i)$a\left(i,i\right)$ is exactly zero; A$A$ is singular and the solution has not been computed.

## Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
For each right-hand side vector b$b$, the computed solution x$x$ is the exact solution of a perturbed system of equations (A + E)x = b$\left(A+E\right)x=b$, where
 |E| ≤ c(k)ε|A| , $|E|≤c(k)ε|A| ,$
c(k)$c\left(k\right)$ is a modest linear function of k$k$, and ε$\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution x$x$ satisfies a forward error bound of the form
 (‖x − x̂‖∞)/(‖x‖∞) ≤ c(k)cond(A,x)ε ,   provided   c(k)cond(A,x)ε < 1 , $‖x-x^‖∞ ‖x‖∞ ≤c(k)cond(A,x)ε , provided c(k)cond(A,x)ε<1 ,$
where cond(A,x) = |A1||A||x| / x$\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }$.
Note that cond(A,x)cond(A) = |A1||A|κ(A)$\mathrm{cond}\left(A,x\right)\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$; cond(A,x)$\mathrm{cond}\left(A,x\right)$ can be much smaller than cond(A)$\mathrm{cond}\left(A\right)$ and it is also possible for cond(AT)$\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ to be much larger (or smaller) than cond(A)$\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_dtbrfs (f07vh), and an estimate for κ(A)${\kappa }_{\infty }\left(A\right)$ can be obtained by calling nag_lapack_dtbcon (f07vg) with norm = 'I'${\mathbf{norm}}=\text{'I'}$.

The total number of floating point operations is approximately 2nkr$2nkr$ if kn$k\ll n$.
The complex analogue of this function is nag_lapack_ztbtrs (f07vs).

## Example

```function nag_lapack_dtbtrs_example
uplo = 'L';
trans = 'N';
diag = 'N';
kd = int64(1);
ab = [-4.16, 4.78, 6.32, 0.16;
-2.25, 5.86, -4.82, 0];
b = [-16.64, -4.16;
-13.78, -16.59;
13.1, -4.94;
-14.14, -9.96];
[bOut, info] = nag_lapack_dtbtrs(uplo, trans, diag, kd, ab, b)
```
```

bOut =

4.0000    1.0000
-1.0000   -3.0000
3.0000    2.0000
2.0000   -2.0000

info =

0

```
```function f07ve_example
uplo = 'L';
trans = 'N';
diag = 'N';
kd = int64(1);
ab = [-4.16, 4.78, 6.32, 0.16;
-2.25, 5.86, -4.82, 0];
b = [-16.64, -4.16;
-13.78, -16.59;
13.1, -4.94;
-14.14, -9.96];
[bOut, info] = f07ve(uplo, trans, diag, kd, ab, b)
```
```

bOut =

4.0000    1.0000
-1.0000   -3.0000
3.0000    2.0000
2.0000   -2.0000

info =

0

```